How do capital structure and economic regime aect fair prices of bank's equity and liabilities?

Donatien Hainaut1 Yang Shen2 Yan Zeng3 ,

,

April 7, 2016

ESC Rennes Business School and CREST, France. 2. York University, Toronto, Canada. 3. Lingnan (University) College, Sun Yat-sen University, Guangzhou 510275, P.R. China. Email: [email protected], [email protected], [email protected] 1.

Abstract This paper considers the capital structure of a bank in a continuous-time regime-switching economy. The modeling framework takes into account various categories of instruments, including equity, contingent convertible debts, straight debts, deposits and deposits insurance. Whereas previous researches concentrate on the determination of the capital structure that maximizes shareholders' equity, this work focuses on the fair pricing of liabilities that ensures no cross-subsidization among stakeholders. This is discussed in a case study where the bank's EBIT is modeled by a four-regime process and is tted to real market data. A numerical analysis reveals that convertible debts can signicantly reduce the cost of deposits insurance and straight debts as well as probabilities of bankruptcy. Although it is found that the risk of dilution for shareholders is important, paradoxically, a high conversion rate for the contingent convertible debt, compensated by a low interest cost before conversion, can delay this dilution. Finally, we nd that in case of change of economic regime, there exists an optimal capital structure from the shareholder's perspective. Keywords : Contingent convertibles, Regime-switching, Wiener-Hopf factorization, Hitting time. JEL Classication: J26; G11 1

Introduction

Banks issue a wide range of liabilities to nance their investments and nowadays bank managers are faced with the task of minimizing the cost of liabilities, while keeping the risk in the balance sheet under control. In addition to traditional liabilities such as deposits and straight debts, a class of new debt instruments, called contingent convertible bonds, has been issued by banks, especially European banks, since the recent crisis. They provide an automatic recapitalization mechanism in case of insolvency, which hence can be used to mitigate the bankruptcy risk. The market for this type of debts was born in December 2009 with the very rst issuance by Lloyds Banking Group. Since then, many other nancial institutions, such as UBS, Credit Suisse and Barclays, have issued contingent convertible debts, which are now commonly called CoCo bonds. The yields of CoCos tend to be higher than those of higher-ranked debt instruments of the same issuer. This paper aims at explaining this phenomenon in a regime-switching economy, by analyzing the relationship between the whole capital structure of the bank and the fair costs of deposits, straight debts and contingent convertible bonds. The optimal capital structure is one of the most important areas in corporate nance, going back to the celebrated MM theorem in Modigliani and Miller (1958). Brennan and Schwartz 1

(1978) provide the rst quantitative examination of optimal leverage. Myers (1984) reviews some of pitfalls encountered when we try to explain the capital structure of rms. Bradley et al. (1984) propose a model that emphasizes the trade o between tax advantages and bankruptcy costs and its role in the choice of a capital structure. Titman and Wessels (1988) analyze empirically the explanatory power of several theories of optimal capital structure. In a serie of papers Leland and Toft (1996) or Leland (1994, 1998) explore dierent ways to optimize endogenously the capital structure. In their works, the rm issues perpetual coupon debt that can be called by the equityholders. The equityholders nd the optimal default barrier by maximizing the rm value. The main argument justifying this approach is that after the debt being issued, the rm is controlled by equityholders. Titman and Tsyplakov (2007) propose a continuous time model of a rm that can dynamically adjust both its capital structure and its investment choices. Hilberink and Rogers (2002) revisit this model and introduce negative jumps in the asset process. Chen and Kou (2009) extend the Leland-Toft model by introducing two sided jumps in the dynamics of rm assets. Banks, as nancial intermediaries, are dierent than other rms. Signicantly, banks have the unique benet of being able to issue federally insured debt; but they also bear the cost of strict capital regulations, including the threat of being placed in receivership and wiping out the investment of the shareholders. The optimal capital structure in the banking industry is considered in Pennacchi (2010), who studies the equilibrium pricing of the bank's deposits, contingent capital and shareholders' equity, when the bank's assets follow a jump-diusion process and the default-free interest rates are stochastic. The work of Koziol and Lawrenz (2012) emphasizes the potential drawbacks of contingent convertibles due to distorted risk incentives. Glasserman and Nouri (2012) derive closed-form expressions for the market value of contingent convertibles when the bank's assets are modeled by a geometric Brownian motion. Barucci and Del Viva (2012) study the optimal capital structure of a bank issuing countercyclical contingent capital. Barucci and Del Viva (2013) extend their previous work to the case with dierent conversion rules. De Spiegeleer and Schoutens (2012, 2013) study CoCo issues with multiple triggers spread across a range of trigger levels. Corcuera et al. (2014) price coupon cancelable CoCo bonds. In line with Ammann and Genser (2005), Hackbarth et al. (2006) or Koziol and Lawrenz (2012), our paper also considers an EBIT-based capital structure model. However, instead of focusing on the maximization of the rm or equity values, our research focuses on the fair valuation of bank liabilities at issuance. In previous cited papers, the equityholders nd the optimal default barrier by maximizing either the rm or the equity values. The main argument justifying this approach is that after the debt being issued, it is the equityholders who control the rm. This optimization criterion does not warranty a fair pricing of debts as the interest rate charge is most of the time assumed exogenous and independent from the capital structure. This is at the origin of a gap between the market value of debts at issuance and the amounts lent by debtholders. For this reason, we opt for an alternative approach in which the fair cost of liabilities is evaluated in a way avoiding cross-subsidization between stakeholders. This allows us next to emphasize the interconnections between the capital structure, cost of deposits, debts, coco bonds and probability of bankruptcy. On the other hand, except Barucci and Del Viva (2012) considering a two-state economy, almost all previous works do not analyze the eect of macroeconomic conditions or business cycles on the market value of bank's liabilities and equity. However, macroeconomic conditions or business cycles do have signicant impacts on the market fundamentals, which should never be ignored in every aspect of nance, such as asset pricing, portfolio selection and risk management. Hamilton (1989) popularizes applications of the so-called regime-switching models in economics and nance. One of the main features of these models is that the parameters in the models are allowed to change over time according to the state of an underlying Markov chain. This provides us with an ideal framework to describe structural changes of macroeconomic conditions or evolutions of business cycles. Indeed, regime-switching models have already received a lot 2

of attention in econometrics, asset pricing and allocation. Guidolin and Timmermann (2005) present empirical evidence of persistent 'bull' and 'bear' regimes in UK stock and bond returns. Similar results are found in Guidolin and Timmermann (2008), for international stock markets. Guidolin and Timmermann (2007) characterize investors' asset allocation decisions under a regime-switching model when asset returns are categorized into four states, namely, crash, slow growth, bull and recovery. Cholette et al. (2009) t skewed-t GARCH marginal distributions for international equity returns and a regime-switching copula. Hainaut and MacGilchrist (2012) study the strategic asset allocation between stocks and bonds when both marginal returns and copula are determined by a hidden Markov chain. Calvet and Fisher (2001, 2004) show that the discredited versions of multifractal processes can capture thick tails and have a regime switching structure with a very large number of states. Elliott et al. (2005) develop a regime-switching Esscher transform and consider the pricing of European options. This paper contributes to the existing literature in several ways. Firstly, it is an extension of Barucci and Del Viva (2012) from a two-regime economy to a multi-regime one. Guidolin and Timmermann (2007) and Gatumel and Ielpo (2011) provide empirical evidences that two regimes are not enough to capture asset returns for many securities and they further point out that a regime economy with two to ve states is required to capture the features of each asset's distribution. This is conrmed by the case study in this paper, in which a regime-switching process with two to four states is tted to real market data. An analysis of loglikelihood, AIC and BIC suggests that the best t is achieved with four economic states. It should be noted that the results of this paper are not limited to a four-state economy, but work for an economy with any nite number of states. Secondly, this research dierentiates with the previous literature in which the discount rate also depends on the economic states. This induces a correlation between the rm's operating prot and the discount rate. Closed-form expressions are established for market values of all bank's liabilities and the deposit insurance. The third contribution is the discussion about the inuence of the capital structure on the fair cost of liabilities, which is a guarantee of no cross-subsidization. Finally, this work proposes a method to retrieve probabilities of CoCo conversion and bankruptcy as well as expected times before these events. We show that convertible debts signicantly reduce the cost of deposits insurance and straight debts. CoCo bonds are indeed assimilable to equity and their presence reduces the probability of bankruptcy and credit spreads. Although it is found that the risk of dilution for shareholders is important, paradoxically, a high conversion rate for the contingent convertible debt, compensated by a low interest cost before conversion, can delay this dilution. When liabilities are issued at fair prices, the shareholders' equity has the same market value, whatever the capital structure of the bank. The switching regime model reveals that this is no more the case if the economic conjuncture changes. From the shareholder's perspective, there exists an optimal structure of debts maximizing the equity value in economic regimes, dierent from the one in force, during the issuance of debts. The rest of this paper is organized as follows. Section 2 introduces the model for the bank's operating prot and investigates the valuation of the bank's asset under a risk neutral measure chosen by the Esscher transform. Section 3 describes dierent categories of liabilities and the deposit insurance. In Section 4, properties of hitting times for a Brownian motion with drift and regime-switching are reviewed. Sections 5 and 6 address the evaluation of nancial instruments in a two-period setting: before and after conversion of the CoCo debts into equity. In Section 7, an inverse Laplace transform approach is introduced to calculate probabilities of conversion and default. The valuation of fair prices of liabilities is presented in Section 8. Section 9 provides a case study to illustrate our results, and Section 10 concludes the paper.

3

2

Bank's earnings and valuation of asset

Any bank's operating prot is aected by structural changes of macroeconomic conditions or market modes, which are usually caused by certain events such as wars, natural catastrophes, terrorist attacks, technological innovations as well as economic booms and recessions. To capture this changing feature, a continuous-time Markov chain is adopted to model the economy and is assumed to modulate the risk free rate and other key parameters in the dynamics of the bank's EBIT. Consider a complete probability space (Ω, F, P) equipped with an augmented ltration F := {Ft }t≥0 generated by a Markov chain and a Brownian motion to be specied below. Here P denotes the real-world probability measure. Throughout this paper, the economy is categorized into N states or regimes, indexed by a set of integers N := {1, 2, · · · , N }. The information about the eective economic state over time1 , is carried by a vector δ(t) taking values from a 0 set of RN -valued unit vectors E = {e1 , . . . , eN }, where ej = (0, . . . , 1, . . . , 0) . The ltration generated by {δ(t)}t≥0 is denoted by {Gt }t≥0 and augmented in the usual way by {Ht }t≥0 . Ht carries the information about the bank's EBIT and is such that Ft = Gt ∨ Ht . The generator of δ(t) is an N × N matrix Q0 := [qi,j ]i,j=1,2,...,N , whose elements satisfy the following standard conditions:

qi,j ≥ 0,

∀i 6= j,

and

N X

∀i ∈ N .

qi,j = 0,

(1)

j=1

Note that when ∆t is small, qi,j ∆t is the approximation probability of a switch from state i to state j , for i 6= j . The matrix of transition probabilities, denoted as P (t, s), is given by

P (t, s) = exp (Q0 (s − t)) ,

s ≥ t,

(2)

and its elements, denoted as pi,j (t, s), i, j ∈ N , are such that

pi,j (t, s) = P (δ(s) = ej | δ(t) = ei ),

i, j ∈ N ,

(3)

where pi,j (t, s) is the transition probability of a switch from state i at time t to state j at time s. The probability of the chain being in state i at time t, denoted by pi (t), depends upon the initial probabilities pk (0) at time t = 0 and the transition probabilities pk,i (0, t), where k = 1, 2, . . . , N , as follows:

pi (t) = P (δ(t) = ei ) =

N X

pk (0)pk,i (0, t),

∀i ∈ N .

(4)

k=1

In Elliott et al. (1994), the following semi-martingale representation theorem for δ(t) is provided ˆ t 0 δ(t) = δ(0) + Q0 δ(s)ds + Mt , (5) 0

RN -valued,

where {Mt }t≥0 is an (G, P)-martingale increment process. Ammann and Genser (2005), Hackbarth et al. (2006) and Koziol and Lawrenz (2012) study an EBIT-based capital structure model. This work adopts the same approach and extends it to the case of a Markov-modulated geometric Brownian motion EBIT dynamics. Specically, the bank's EBIT, also used as an approximation of the bank's cash-ow, is assumed to be eXt at time t and the dynamics of {Xt }t≥0 under P is governed by the following Markov-modulated Brownian motion with drift:

dXt = µ ¯t dt + σt dWt , 1

X0 = x0 ,

(6)

Notice that the economic regime is assumed observable. However, as explained in the last paragraph of section 6, most of our results may be extended to hidden regimes.

4

where {Wt }t≥0 is an (F, P)-standard Brownian motion. Here the drift and volatility parameters are modulated by the Markov chain. That is, µ ¯t = δ(t)0 µ ¯ and σt = δ(t)0 σ , where 0 0 µ ¯ = (¯ µ1 , . . . , µ ¯N ) and σ = (σ1 , . . . , σN ) are the vectors of drifts and
volatilities in all eco2 0 denote the vector of nomic regimes. With a little abuse of notation, let σ 2 = σ12 , . . . , σN squared volatilities in all economic regimes. Macroeconomic conditions also inuence the risk-free rate. As observed in Japan over the last decade, or in Europe recently, economic recessions can cause deation and low interest rates, while economic booms can result in ination and high interest rates. To integrate this feature in the modeling framework, the risk-free rate is assumed to be determined by the economic regime 0 as rt = δ(t)0 r, where r = (r1 , . . . , rN ) denotes the vector of risk-free rates in all economic regimes. The next proposition recalls an useful result established by Elliott and Siu (2013), which is used throughout this paper.

Proposition 2.1. Let B(u) be an N × N diagonal matrix of φ(u) = µ¯u + 21 u2 σ2 , i.e. diag (φ(u)). The Laplace transform of Xt under P is given by:   0 EP euXt = eux0 δ(0) exp ((Q0 + B(u)) t) 1,

B(u) = (7)

where 1 is a vector of ones and EP [·] denotes the expectation under P. Before introducing the capital structure, the remainder of this section focuses on the valuation of bank's asset. The totality of bank's cash-ow 2 is generated by the asset, which is traded in nancial markets. To avoid arbitrage opportunities, the price of the asset should be equal to the expected present value of the future net cash ow under a risk neutral measure Q. However, as is known, the nancial market with regime-switching is incomplete by nature and there exists more than one risk neutral measures. Under dierent criteria, dierent equivalent measures can be suggested. For example, the minimal entropy martingale measure and the variance-optimal martingale measure are constructed by minimizing the Kullback-Leibler distance, i.e. the relative entropy, and the L2 -norm, respectively. Although the selection of equivalent measures itself is an important research topic in nance, this paper will not discuss which criterion is better. Instead, the Esscher transformation method is adopted directly to select the risk neutral measure. The Esscher transform is a time-honored tool in actuarial science and has been promoted by Gerber and Shiu (1994) in option pricing. The merit of the risk neutral measure determined by the Esscher tranform is that it provides a general, transparent and unambiguous framework. As shown in Elliott et al. (2005), a regime-switching Esscher transform is dened by a so-called regime-switching Esscher parameter ξt = δ(t)0 ξ , where ξ = (ξ1 , ξ2 , ..., ξN )0 . Specically, the risk neutral measure Q is equivalent to P through the Radon Nykodym density as follows ´t ´ ´t 1 t 2 2 dQ e 0 ξs dXs h ´t i = e− 2 0 ξs σs ds+ 0 ξs σs dWs . = dP Ft EP e 0 ξs dXs | Gt

Since ξ and σ are constant and hence bounded, the stochastic exponential on the right hand side of the above equation is an (F, P)-martingale and then the equivalent measure Q is well dened. The method to infer the value of ξ is explained at the end of this section. The next proposition gives the dynamics of {Xt }t≥0 under Q, where its drift is modied.

Proposition 2.2. The process {Xt }t≥0 is governed by the following SDE under Q dXt = µt dt + σt dWtQ , 2

X0 = x0 ,

In the model, the EBIT is used as an estimate of the rm's cash-ow.

5

(8)

where

ˆ WtQ = Wt −

t

ξs σs ds,

(9)

∀t ≥ 0,

0

is a standard Brownian motion under Q. Here the drift of Xt is equal to µt = δ(t)0 µ with the vector µ dened by µ= µ ¯j + ξj σj2


0 j=1,...,N

(10)

.

Proof. From Girsanov's theorem, the process dened by (9) is a standard Brownian motion under Q. Substituting (9) into (6) immediately gives the dynamics of Xt under Q and its drift, i.e. (8) and (10). At time t, the value of the bank's asset, denoted by At , is the expected present value of future net cash ow under the risk neutral measure Q. If the tax rate is denoted by γ , At is equal to the following expression   ˆ +∞ ´ − ts ru du Xs (11) At = E (1 − γ) e e ds | Ft . t

Here and afterward E[·] denotes the expectation under Q. This value should be seen as the purchase price of assets under the bank's management. Its closed-form expression is detailed in the next proposition, which is proportional to the bank's current EBIT:

Proposition 2.3. The market values of the bank's asset At , is equal to:  −1   0 1 , At = −(1 − γ)ext δ(t) Q0 + B ξ

(12)

where "

ξ

B = diag

under conditions

µ ¯j +

ξj σj2

1 + σj2 − rj 2

1 µ ¯j + ξj σj2 + σj2 < rj , 2

#

0

(13)

j=1,...,N

j = 1, ..., N .

(14)

Proof. These results are obtained by direct integration and using a variant of Proposition 2.1: ˆ

∞

At = (1 − γ)ext ˆt ∞ = (1 − γ)e

xt

h ´s i ´s 1 2 2 EP e t (µu −ru − 2 ξu σu )du+ t (1+ξu )σu dWu | Ft ds 0



ξ





δ(t) exp Q0 + B (s − t) 1ds t  −1    s=∞ 0 xt ξ ξ exp Q0 + B (s − t) 1 = (1 − γ)e δ(t) Q0 + B s=t   −1  0 = −(1 − γ)ext δ(t) Q0 + B ξ 1 .

If conditions given by (14) are violated, the asset value is innite. Eq. (12) reveals that the asset value At is not continuous. As it depends upon δ(t) directly, a change of economic regime causes a sudden jump in the asset value. The next proposition introduces additional conditions that ξ fullls to guarantee the absence of arbitrage. 6

Proposition 2.4. Vector ξ = (ξ1 , ξ2 , ..., ξN ) denes a risk neutral measure if and only if it is the solution of the following system: ri = µ ¯i +

ξi σi2

0


−1

0

−1

e j Q0 + B ξ

X 1 + σi2 + qi,j 2

ei (Q0 + B ξ )

j6=i

1 1

! −1 ,

i = 1, ..., N.

(15)

Proof. Using Itô's lemma for semi-martingales leads to the following dynamics for the asset: 1 dAt = At dXt + At hdXt , dXt i + (At+ − At ) dδ(t). 2

(16)

(i)

Then, if δ(t) = ei and the asset value in this state is noted by At , the expectation of Eq. (16) is hence given by !   (j) X 1 2 At (i) (i) µi + σi dt + − 1 At dt, (17) E [dAt | Ft ] = At qi,j (i) 2 A j6=i

where the ratio

(j)

At

(i)

At

t

is independent from time and equal to (j)

At

(i)

At

=

0


−1

0

−1

ej Q0 + B ξ

ei (Q0 + B ξ )

1 1

.

Since expectation (17) under the risk neutral measure must be equal to the risk free rate in (i) regime i, i.e. E [dAt | Ft ] = ri At dt, system (15) immediately follows. In numerical applications, the system of Eqs. (15) is solved numerically so as to infer ξ . The next section introduces dierent categories of liabilities, which are used to nance the purchase of the bank's asset. Note that all further developments are done under the risk neutral measure Q. 3

Bank's equity and liabilities

Suppose that the bank is participating in nancial markets to fulll its nancing needs through issuing deposits, straight bonds and CoCo bonds. Deposits and straight bonds are assumed to be xed coupon consols, paying continuous coupons at the rates of π1d and π1sd , respectively. The total charge of these consol bonds is denoted by π1 = π1d + π1sd . Deposits are insured in case of bankruptcy and the bank pays a fair deposit insurance premium in exchange, denoted as DIt . In case of bankruptcy, assets owned by the bank are sold at a signicant discount and used solely to repay holders of deposits and straight bonds. To enhance the solvency, the bank also issues CoCo bonds. Before conversion, these debts are similar to straight debts and pay a continuous coupon at the rate of π2 . In case of insolvency, they are converted into equity and the payment of coupons is terminated. Under the assumption that the total earnings are distributed to shareholders, if γ is the tax rate, a total dividend of (1 − γ)(eXt − π1 − π2 ) is paid before the conversion of CoCo bonds. If CoCo bonds are swapped into equity, the total dividend distributed to all shareholders, including former CoCo bondholders, becomes (1 − γ)(eXt − π1 ). The number of existing shares and potential new shares are denoted by NS and NC , respectively. Then upon conversion, earnings S allocated to former shareholders and new shareholders are thus NSN+N (1 − γ)(eXt − π1 ) and C NC Xt − π ), respectively. 1 NS +NC (1 − γ)(e 7

According to Basel II and III, insolvency is triggered when the Mc Donough ratio (Tier I and II core equity on Total Risk Weighted Assets) falls below 8%. In this model, the event of nancial distress is instead approached by a lower bound on the EBIT. The conversion of CoCo bonds to equity is triggered by the event that the EBIT falls below a threshold proportional to the sum of payments to all debtholders, i.e., θ(π1 + π2 ), where θ is either chosen by the regulator or shareholders. Then the stopping time corresponding to the conversion is dened as follows: (18)

τ1 = inf {t ≥ 0 : Xt < ln (θ (π1 + π2 ))} .

Once CoCo bonds have been swapped, the bank has no other tool to mitigate the risk of insolvency. The bank enters bankruptcy when the EBIT falls below the same regulatory threshold. After conversion, the default time is then dened by (19)

τ2 = inf {t ≥ 0 : Xt < ln (θπ1 )} .

At any time t before conversion, the market value of bank equity, gross of deposit insurance 3 , denoted by St , is the sum of two terms:   ˆ τ1 ´ ´τ
NS − ts ru du Xs − t 1 ru du e Sτ1 | Ft , t < τ1 , (20) St = E e (1 − γ) e − π1 − π2 ds + NS + NC t where the rst term is the expected discounted value of future dividends till the conversion of CoCo bonds into equity, and the second term is the part of capital owned by the incumbent (existing) equityholders. Here Sτ1 is the total bank equity at the instant that CoCo bonds is swapped into new shares. At time τ1 ≤ t ≤ τ2 , this market value is the sum of discounted cash ows till default and the recovery value of bank's asset upon default: ˆ τ2 ´  ˆ ∞ ´
− ts ru du Xs − ts ru du Xs St = E e (1 − γ) e − π1 ds + (1 − λ)(1 − γ) e e ds | Ft , (21) t

τ2

where 1−λ is the (constant) recovery rate of bank's assets, after liquidation, transferred to shareholders. In this framework, equity values per share before and after conversion are respectively t given by NSSt if t < τ1 and NSS+N if τ1 ≤ t ≤ τ2 . C Straight debts and deposits are both modeled as consol bonds and their market values are the expected discounted sum of future coupons, net of tax, and of recovered principals in case of bankruptcy. Under the assumption that the residual value is directly proportional to interests paid, their market values can be respectively represented as follow for t ≤ τ2 :  ˆ τ2 ´  ˆ ∞ ´ π1sd 1,sd − ts ru du sd − ts ru du Xs e (1 − γ)π1 ds + λ Dt = E (1 − γ) e e ds | Ft , (22) π1 t τ2 and

Dt1,d

ˆ =E

τ2

e t

−

´s t

ru du

(1 −

γ)π1d ds

πd + λ 1 (1 − γ) π1

ˆ

∞

e

−

´s t

ru du Xs

e

 ds | Ft .

(23)

τ2

The insurance purchased by the bank for hedging deposits is a put option on their market values (without tax benet given that the insurance is triggered only in case of bankruptcy). The strike price of this option is the residual value of bank's assets allocated to depositors. Under the assumption that this residual value is directly proportional to interests paid, the fair insurance premium is equal to  ´   ˆ ∞ ´  τ − τs ru du − t 2 ru du d DIt = E e E π1 e 2 ds | Fτ2 τ2 #   d ˆ ∞ ´s π1 − τ ru du Xs − E λ (1 − γ) e 2 e ds | Fτ2 | Ft . (24) π1 τ2 + 3

The equity net of the deposit insurance is dened later as the equity value decreased by the premium for the deposit insurance.

8

On the other hand, the market value of CoCo bonds is the sum of coupons paid till conversion and the capital after the swap of these bonds into equity:  ˆ τ1 ´ ´τ NC − t 1 ru du 2 − ts ru du (25) Sτ1 | Ft . e (1 − γ)π2 ds + Dt = E e NS + NC t C After conversion, for any time t ≥ τ1 , former CoCo bondholders own NSN+N St of equity. In C addition, the rm's value is the sum of equity, deposits, straight bonds and CoCo bonds and minus the insurance premium. It is denoted by Vt and has the following expression   ˆ ∞ ´ − ts ru du Xt (26) e ds | Ft − DIt . e Vt = E (1 − γ)

t

It is interesting to note that the rm's value does not depend upon the conversion time of CoCo bonds into equity, but only depends on the default time, which determines the insurance premium. All results can be extended to a stochastic recovery rate, independent from the economic regime. Fair values of liabilities and equity are still computable by propositions 5.1 to 6.2 that follows later. But in this case, the constant λ has to be replaced by its average E(λ). This immediate extension is due to the fact that prices are expectations. The introduction of stochastic recovery rate has however an impact on the risk borne by debtholders. This is conrmed by Altman et al. (2002), who show that random recovery rates raise signicantly the value at risk of debts but has a limited impact on provisions for credit losses. Notice that the recovery rate can eventually be linked to the economic regime. This does not present any technical diculty but requires to replace in propositions 5.1 to 6.2 the product λ1 by a vector Λ that contains the expected recovery rates in each economic state. 4

Properties of hitting times

The solution of the rst passage problem for (δ(t), Xt ) across a constant level is related to the down-crossing ladder process Xt := min0≤s≤t Xs . Indeed, Xt is also a Markov process on the same state space and its generator matrix is related to the matrix Wiener-Hopf factors (Q+ , Q− ) of (δ(t), Xt ).

Denition 4.1.

A pair of irreducible N × N matrices (Qr,+ , Qr,− ), i.e. matrices with nonnegative o-diagonal entries and non-positive row sums, is called the Wiener-Hopf factorization of (δ(t), Xt ) associated with r > 0 if the following second-order matrix-valued equations hold, i.e.

Ξ(−Qr,+ ) = Ξ(Qr,− ) = 0,

(27)

1 Ξ(Qr ) := Σ2 Qr 2 + V Qr + Q0 − R, 2

(28)

where

with Σ = diag(σ), V = diag(µ) and R = diag(r). Rogers (1994) proposes an algorithm to compute the matrix Wiener-Hopf factors by diagonalization. Jiang and Pistorius (2008) prove the uniqueness of the matrix Wiener-Hopf factors and establish an analytical expression for the Laplace transform of a hitting time in a more general jump-diusion framework. The next proposition is a direct consequence of this last result and gives the relationship between the rst passage time and the matrix Wiener-Hopf factors of (δ(t), Xt ). 9

Proposition 4.2. Consider the rst passage time τ of Xt below a constant level β , τ = inf {t ≥ 0 : Xt < β} ,

and the contingent payo h(τ ) depends on the economic regime as h(τ ) = δ(τ )0 h, where h = (h1 , ..., hN )0 . The expected discounted value of the contingent payo at time τ is equal to h ´τ i 0 E e− 0 rs ds h(τ ) | F0 = δ(0) exp (Qr,− (x0 − β)) h,

(29)

where Qr,− is the Wiener-Hopf factor determined by Eqs. (27)-(28). The following algorithm is proposed by Rogers (1994) to derive the Wiener-Hopf factors from the second-order matrix-valued equations (27)-(28):

• Step 1: Calculate pairs of eigenvalues and eigenvectors (βi,u , vi ) ∈ (C, CN ), for i = 1, 2, . . . , 2N , of      0 I0 vi vi = , 2Σ−2 (R − Q0 ) −2Σ−2 V wi wi where I0 is an N × N matrix.

• Step 2: Sort pairs of eigenvalues and eigenvectors according to real parts of eigenvalues (Re(β1,u ) ≤ Re(β2,u ) ≤ . . . ≤ Re(βN,u )). • Step 3: Set Z− := [v1 , . . . , vN ] and Z+ := [vN +1 , . . . , v2N ], then Qr,− is −1 Q− = Z− diag (β1,u , ..., βN,u ) Z+ ,

and its matrix exponential can be computed as   −1 . exp (Q− x) = Z− diag eβ1,u x , . . . , eβN,u x Z+ Proposition 4.2 and the above algorithm will be used in the following sections to derive analytical expressions for the bank's equity and liabilities, including deposits, straight bonds and CoCo bonds as well as the insurance premium. 5

τ ≤

Valuation of bank's equity and liabilities after conversion ( 1

t < τ2 ) This section focuses on the period after the swap of CoCo bonds against equity. During this period, the bank is leveraged by deposits and straight debts and there remain only three claimants: depositors, debtholders and shareholders. However by denition, the market values of deposits, straight debts and insurance are independent from the conversion time of CoCo bonds. Then, their expressions also hold for any time before conversion.

Proposition 5.1. The market value of straight debts before default is equal to 0

Dt1,sd = (1 − γ)π1sd δ(t) [I0 − exp (Qr,− (xt − ln (θπ1 )))] (R − Q0 )−1 1 0

−1

+(1 − γ)λπ1sd θδ(t) exp (Qr,− (xt − ln (θπ1 ))) (R − B − Q0 )

where B = B(1) is dened in Proposition 2.1.

10

(30)

1,

t < τ2 ,

Proof. By denition, the market price of straight debts is the sum of two terms: Dt1,sd

= (1 −

γ)π1sd E

ˆ

τ2

e

−

´s t

ru du

 ds | Ft

t

π sd + (1 − γ)λ 1 E π1

ˆ

∞

e

−

´s t

ru du Xs

e

 ds | Ft .

(31)

τ2

The rst term of Eq. (31) can be decomposed into the dierence of two expectations as follows    ´  ˆ ∞ ´  ˆ τ2 ´ ˆ ∞ ´s τ − τ ru du − t 2 ru du − ts ru du − ts ru du 2 e ds | Ft , (32) e ds | Ft − E e E e ds | Ft = E τ2

t

t

and from Proposition 2.1, the rst expectation in the above expression can be calculated as follows  ˆ ∞ ´ ˆ ∞ h ´ i s − ts ru du E e− t ru du | Ft ds E e ds | Ft = t ˆt ∞ 0 δ(t) exp ((Q0 − R) (s − t)) 1ds = ht 0 is=∞ = δ(t) (Q0 − R)−1 exp ((Q0 − R) (s − t)) 1 s=t

0

= δ(t) (R − Q0 )−1 1.

(33)

Using the tower property of conditional expectations to the second expectation on the right hand side of (32) gives  ´   ´ ˆ ∞ ´   ˆ ∞ ´s τ2 τ2 − r du − s r du E e− t ru du e τ2 u ds | Ft = E e− t ru du E e τ2 u ds | Fτ2 | Ft τ2 τ  ´  ˆ ∞2 τ2 0 − t ru du = E e δ(τ2 ) exp ((Q0 − R) (s − τ2 )) 1ds | Ft τ2   i h ´ τ2 0 = E e− t ru du δ(τ2 ) (R − Q0 )−1 1 | Ft 0

= δ(t) exp (Qr,− (xt − ln (θπ1 ))) (R − Q0 )−1 1.

(34)

where the last equality follows from Proposition 4.2. Combining Eqs. (33) and (34) yields ˆ τ2 ´  0 − ts ru du E e ds | Ft = δ(t) [I0 − exp (Qr,− (xt − ln (θπ1 )))] (R − Q0 )−1 1. (35) t

In the same vein, using the tower property of conditional expectations to the second term in (31) leads to ˆ ∞ ´   ˆ ∞ ´   − ts ru du Xs − ts ru du Xs E e e ds | Ft = E E e e ds | Fτ2 | Ft τ2 τ2  ´  ˆ ∞ h ´s i τ − τ ru du Xt −Xτ − t 2 ru du ln(θπ1 ) 2 2 = E e e E e e | Fτ2 ds | Ft τ2 h ´ τ2 i 0 −1 − t ru du ln(θπ1 ) = E e e δ(τ2 ) (R − B − Q0 ) 1 | Ft 0

= eln(θπ1 ) δ(t) exp (Qr,− (xt − ln (θπ1 ))) (R − B − Q0 )−1 1 0

= θπ1 δ(t) exp (Qr,− (xt − ln (θπ1 ))) (R − B − Q0 )−1 1.

(36)

Combining (35) and (36) completes the proof. The fair consol interest payment π1sd should be determined at the issuance of straight debts, such that the market value of straight debts is equal to the amount brought by bondholders. This ensures the absence of arbitrage. However as shown by Eq. (30), this fair price is directly related to the total interest π1 paid to depositors and debtholders, and then to the whole capital structure of the bank. This point will be further discussed in Section 8. The next proposition introduces an analytical expression for price of the deposits insurance. 11

Proposition 5.2. The value of the fair insurance premium before default is equal to 0

DIt = δ(t) exp (Qr,− (xt − ln(θπ1 ))) c,

(37)

t < τ2 ,

where c = (c1 , c2 , . . . , cN ) is the vector value of claims covered by the deposits insurance in all economic regimes with 0

  ci = π1d e0i (R − Q0 )−1 1 − λ(1 − γ)π1d θe0i (R − B − Q0 )−1 1

(38)

+

for i = 1, . . . , N . Proof. As proved in Proposition 5.1, the market value of deposits when the default event occurs is such that

 ˆ E π1d

∞

−

e

´s τ2

ru du

 0 ds | Fτ2 = π1d δ(τ2 ) (R − Q0 )−1 1,

(39)

τ2

and the market value of assets sold is equal to  ˆ ∞ ´ 0 − τs ru du Xs 2 E e ds | Fτ2 = eln(θπ1 ) δ(τ2 ) (R − B − Q0 )−1 1. e

(40)

τ2

Combining expressions (39) and (40) gives the expressions for ci . Then a direct application of Proposition 4.2 gives the fair price of the insurance. In this paper, deposits are like consol bonds except that they are hedged by an insurance contract. As stated in the next proposition, their price has almost the same expression as straight debts. Given that their value is closely related to π1 , the fair interest paid to depositors is then directly inuenced by the size of straight debts. On the other hand, interests paid to depositors should be smaller than those paid to debtholders, because the deposits insurance reduces their exposure to bankruptcy. The relationships between the fair deposits rate, volume of straight debts and insurance fee will be developed in Section 8.

Proposition 5.3. The market value of deposit before default is equal to 0

Dt1,d = (1 − γ)π1d δ(t) [I0 − exp (Qr,− (xt − ln (θπ1 )))] (R − Q0 )−1 1 0

(41)

+(1 − γ)λπ1d θδ(t) exp (Qr,− (xt − ln (θπ1 ))) (R − B − Q0 )−1 1,

t < τ2 ,

where B = B(1) is dened in Proposition 2.1. The proof of Proposition 5.3 is the same as that of Proposition 5.1. Contrary to deposits, straight debts or insurance premium, the expression for the market value of equity diers before and after conversion of CoCo bonds. After conversion, former shareholders are diluted and the total equity value is the sum of all expected discounted dividends.

Proposition 5.4. Given that Xt = xt , the equity value St after conversion and gross of deposit insurance is equal to h i 0 0 St = (1 − γ) ext δ(t) I0 − λθπ1 δ(t) exp (Qr,− (xt − ln (θπ1 ))) (R − B − Q0 )−1 1 0

−(1 − γ)π1 δ(t) [I0 − exp (Qr,− (xt − ln (θπ1 )))] (R − Q0 )−1 1,

where B = B(1) is dened in Proposition 2.1.

12

τ1 ≤ t < τ2 ,

(42)

Proof. By denition, the equity value can be decomposed as follows ˆ

∞

−

´s

ˆ



ru du Xs

ds | Ft − λ(1 − γ)E  τ2 ´s e− t ru du ds | Ft .

St = (1 − γ)E ˆ −π1 E

e

t

∞

e

e

t

−

´s t

ru du Xs

e

 ds | Ft

τ2

(43)

t

The rst expectation in Eq. (43) can be derived as  ˆ ∞ ´ ˆ ∞ h ´ i s s E e− t ru du eXs | Ft ds E e− t ru du eXs ds | Ft = t ˆt ∞ 0 ext δ(t) exp ((Q0 + B − R) (s − t)) 1ds = t xt

0

= e δ(t) (R − B − Q0 )−1 1. The second and the third expectations in Eq. (43) can be derived as Eqs. (36) and (32) in Proposition 5.1. Combining them completes the proof. After the swap of CoCo bonds into equity, the rm's value is the sum of equity, straight debts and deposits, minus the insurance premium (respectively provided by Eqs. (30), (37), (41) and (42)). A direct calculation leads to the following corollary.

Corollary 5.5. The rm's value Vt = St +Dt1,sd +Dt1,d −DIt after conversion and before default is given by (44)

Vt = At − DIt xt

0

−1

= (1 − γ)e δ(t) I0 (R − B − Q0 ) 6

0

1 − δ(t) exp (Qr,− (xt − ln(θπ1 ))) c,

τ1 ≤ t < τ2 .

t < τ1 )

Value of CoCo bonds and equity before conversion (

Before conversion, there are four claimants: shareholders, contingent capital holders, straight debtholders and depositors. Deposits and straight debts are independent of conversion and their market values can be calculated using the same formulas given in Propositions 5.1 and 5.2. The value of CoCo bonds, as shown in the next proposition, is the sum of two terms: one related to coupons paid till conversion and the other related to swapped equity in case of insolvency.

Proposition 6.1. The market value of CoCo bonds before conversion is given by 0

Dt2 = (1 − γ)π2 δ(t) [I0 − exp (Qr,− (xt − ln (θ (π1 + π2 ))))] (R − Q0 )−1 1 NC 0 + δ(t) exp (Qr,− (xt − ln (θ (π1 + π2 )))) SC , t < τ1 , NS + NC

(45)

where SC is the vector of equity value after conversion of CoCo bonds into equity for all economic regimes:     π2 SC = (1 − γ) θ(π1 + π2 )I0 − λθπ1 exp Qr,− ln 1 + (R − B − Q0 )−1 1 π1     π2 −(1 − γ)π1 I0 − exp Qr,− ln 1 + (R − Q0 )−1 1. (46) π1

Proof. By denition, the market value of CoCo bonds can be decomposed as follows: Dt2

ˆ

= (1 − γ)π2 E

τ1

e

−

´s t

ru du

 ds | Ft +

t

13

h ´ τ1 i NC E e− t ru du Sτ1 | Ft . NS + NC

(47)

An analytical expression for the rst term of Eq. (47) can be derived as Eq. (35) in the proof of Proposition 5.1. At time τ1 , xτ1 = ln (θ (π1 + π2 )) and Sτ1 is just a function of the state variable such that     π2 0 0 Sτ1 = (1 − γ) θ(π1 + π2 )δ(τ1 ) I0 − λθπ1 δ(τ1 ) exp Qr,− ln 1 + (R − B − Q0 )−1 1 π1     π2 0 −(1 − γ)π1 δ(τ1 ) I0 − exp Qr,− ln 1 + (R − Q0 )−1 1. π1 According to Proposition 4.2, the second expectation in Eq. (47) can thus be developed as follows: i h ´ τ1 0 (48) E e− t ru du Sτ1 | Ft = δ(t) exp (Qr,− (xt − ln (θ (π1 + π2 )))) SC .

The fair CoCo interest π2 should be determined at the issuance of convertible debts, such that their market value is equal to the amount brought by CoCo bondholders to exclude any arbitrage opportunity. But as shown by Eq. (45), this fair rate is directly related to interests paid to holders of other liabilities. This interdependence is illustrated in Section 8. Using a similar approach as for CoCo bonds leads to a closed-form formula for the equity before conversion:

Proposition 6.2. The market value of the equity before conversion is given by i h 0 0 St = (1 − γ) ext δ(t) − θ(π1 + π2 )δ(t) exp (Qr,− (xt − ln (θ (π1 + π2 )))) (R − B − Q0 )−1 1 0

−(1 − γ) (π1 + π2 ) δ(t) [I0 − exp (Qr,− (xt − ln (θ (π1 + π2 ))))] (R − Q0 )−1 1 NS 0 + δ(t) exp (Qr,− (xt − ln (θ (π1 + π2 )))) SC , t < τ1 , NS + NC

(49)

where SC is dened by equation (46). Proof. Remember that at any time t < τ1 , St is equal to the following sum: ˆ St = (1 − γ)E

τ1

e

−

´s t

ru du Xt

ˆ



e ds | Ft − (1 − γ)E

t

+

τ1

−

e

´s t

ru du

 (π1 + π2 ) ds | Ft

t

h ´ τ1 i NS E e− t ru du Sτ1 | Ft . NS + NC

(50)

As shown in the proof of Proposition 5.4, the expected sums of the discounted EBIT and discounted coupons are respectively given by the following two equations  ˆ τ1 ´  h i s 0 0 E e− t ru du eXt ds | Ft = ext δ(t) I0 − θ(π1 + π2 )δ(t) exp (Qr,− (xt − ln (θ (π1 + π2 )))) t

× (R − Q0 − B)−1 1, and ˆ E

τ1

e

−

´s t

ru du

 (π1 + π2 ) ds | Ft

0

= (π1 + π2 ) δ(t)

(51)

(52)

t

× [I0 − exp (Qr,− (xt − ln (θ (π1 + π2 ))))] (R − Q0 )−1 1. Combining the above two equations with Eqs. (48) and (50) leads to the desired result.

14

The rm's value Vt is the sum of Eqs. (30), (41), (45) and (49), minus the insurance expressed by (37), i.e. Vt = St1 + Dt2 + Dt1,d + Dt1,sd − DIt . It is easy to check that the expression of the rm's value for any time before conversion is equal to that after conversion and before default. That is, (53)

Vt = At − DIt 0

xt

−1

= (1 − γ)e δ(t) I0 (R − B − Q0 )

1

0

−δ(t) exp (Qr,− (xt − ln(θπ1 ))) c,

t < τ1 .

Remark:

in this work, the Markov chain dening the economic regime is observable. The ltration F is then the augmentation of G , the ltration of δ(t), and of H, the ltration of the EBIT process. The main purpose of this assumption is to alleviate the presentation of mathematical developments. In numerical illustrations, the current economic regime can be set for this reason to the most likely one, ltered by the Hamilton algorithm (1989). However our results are extendable to the case of a hidden Markov chain. In this case, fair values of assets, liabilities and equity are equal to the expectation of their equivalents on the enlarged ltration, conditioned by the reduced ltration H. In practice, this expectation is simply the sum of fair values in each regime, weighted by the probability of presence in this regime. For example, let δ(t) us momentarily denote by St the equity value if the economy is in the regime δ(t), as dened in proposition 6.2. The equity value whether δ(t) is not visible is equal to the sum: δ(t)

St = E(St

| Ht ) =

N X

e

pj (t)St j

j=1

The probabilities of presence are again computable with the Hamilton lter, applied directly to the earning process or to a related quantity. The next section develops some interesting indicators for risk management. 7

Expected times and probabilities before conversion and ruin

Introducing CoCo bonds to a bank's balance sheet is an ecient way to mitigate the risk of bankruptcy. However, former major shareholders bear the risk of being diluted in case of insolvency when CoCo bonds are swapped into equity, and losing control of the rm. This section introduces several indicators that can be used to monitor the risk of bankruptcy or dilution. The rst useful tool is the expected lifetime before conversion or bankruptcy. The second tool is the probability of ruin or conversion. Both expected lifetime and probability are retrieved numerically by deriving the characteristic functions of conversion and bankruptcy times. The framework is applied here under the pricing measure Q, but it is also applicable under the real world probability measure P. In the latter case, it suces to replace the parameters of Xt by those observed under P. If β1 and β2 respectively denotes θ(π1 + π2 ) and θπ1 , the characteristic functions of τk for k = 1, 2 are dened by
ϕk (u) = E eiuτk | F0
0 = δ(0) exp Q(−iu),− (x0 − ln (βk )) 1, k = 1, 2, (54) where Q(−iu),− are solutions of the following second-order matrix-valued equations

1 2 2 Σ Q(−iu),− + V Q(−iu) + Q0 + iuI0 = 0, 2 Then, the n-th moments of τk for k = 1, 2 are given by n n n ∂ E (τk | F0 ) = (−i) ϕ (u) , k ∂un u=0 15

i = 1, 2, · · · , N.

k = 1, 2.

(55)

∂n Since there is no closed-form expression for ∂u n ϕ(iu) u=0 , the rst and second order derivatives are computed numerically. The capital structure also determines probabilities of default and conversion of CoCo bonds into equity over a certain time horizon. These probabilities can be retrieved from the Laplace transform of the hitting time τk , for k = 1, 2. By denition, for a given constant α, the Laplace transform of τk is such that

0 (56) E e−ατk | F0 = δ(0) exp Q(α),− (x0 − ln (βk )) 1 ˆ +∞ e−αs P (τk ≤ s | F0 )ds = α 0

= αLα (P (τk ≤ s | F0 )),

k = 1, 2,

where Lα is the Laplace operator. The probability of conversion or default is then obtained by inverting the Laplace transform as follows    1  −ατk P (τk ≤ s | F0 ) = L−1 E e | F 0 α α ˆ γ+iT
1 1 eαs E e−ατk | F0 dα, lim = 2πi T →∞ γ−iT α where γ is greater than the real part of all singularities of E (e−ατk | F0 ). In numerical applications, this transform is inverted using Talbot's method such as detailed in Abate and Whitt (2006). 8

Fair pricing

Table 1 presents the economic balance sheet at time t, in which market values of all items are accounted. As the premium of the deposit insurance is an immediate expense for the bank, it is deducted from both asset and equity. The equity, net of deposits insurance, is denoted by StI = St − DIt . The remainder of this section introduces the conditions that ensure no cross-subsidization among deposits, straight debts and CoCo bonds. Table 1: Economic balance sheet of the nancial institution at time t Assets Equity and liabilities Equity, net of insurance StI = St − DIt Assets, CoCo bonds Dt2 At −DIt net of insurance Straight debts Dt1,sd Deposits Dt1,d Firm Value Vt =At −DIt Firm Value Vt = St + Dt2 + Dt1,sd + Dt1,d − DIt In order to clearly emphasize the link between the capital structure and costs of liabilities, it is assumed that the bank is founded at time t = 0. The amounts of cash invested by equityholders, CoCo-bondholders, straight-bondholders or depositors are respectively denoted by E0 , CCD0 , SD0 and CD0 . The total funds raised by the bank is invested in an asset such that A0 = E0 + CCD0 + SD0 + CD0 and the dynamics of cash-ows Xt paid by this asset is such that relationship (11) is satised. To exclude arbitrage opportunities, coupons paid to the three types of debts must guarantee that the market values of equity and liabilities are equal to the cash invested exactly:   E0 = S0I = S0 − DI0 ,    CCD = D2 , 0 0 (57) 1,sd SD0 = D  0 ,   CD = D01,d + DI0 . 0 16

The last equation of system (57) guarantees that there is no subsidization of deposits insurance by equity holders or by CoCo bondholders and straight bondholders. If such an equation is satised, the insurance premium is only nanced by deposits. On the other hand, the rm's value is equal to F0 = A0 − DI0 . Assume that the recovery rate λ, the default trigger θ and numbers of stocks NS or NC are known. The next corollary introduces a system of equations to determine the fair coupons π1d and π1sd .

Corollary 8.1. If straight bondholders and depositors respectively bring CSD0 and CD0 in cash, the fair coupons are solutions of the following nonlinear system of equations:   π1sd         π1d     

h 0 δ(0) [I0 − exp (Qr,− (x0 − ln (θπ1 )))] (R − Q0 )−1 1 i−1 0 +λθδ(0) exp (Qr,− (x0 − ln (θπ1 ))) (R − B − Q0 )−1 1 , h 0 (CD0 −DI0 (π1d )) = δ(0) [I0 − exp (Qr,− (x0 − ln (θπ1 )))] (R − Q0 )−1 1 (1−γ) i 0 λθδ(0) exp (Qr,− (x0 − ln (θπ1 ))) (R − B − Q0 )−1 1 ,

=

CSD0 (1−γ)

(58)

where DI0 (π1d ) is the value of the deposit insurance calculated by equation (37) and π1 = π1sd +π1d . π1sd CSD0 D01,sd and

From the above corollary, the fair yields for depositors and bondholders are thus y1sd = π1d sd d CD0 . By construction, when π1 and π1 satisfy system 1,d D0 + DI0 . The money invested by depositors is then equal to

and y1d =

(58), SD0 =

the sum of market values CD0 = of deposits and deposit insurance fee. System (58) is easily solved numerically in an example presented in the next section. The next result presents the equation that the fair coupon π2 has to satisfy.

Corollary 8.2. If CoCo bondholders invest an amount of the following nonlinear equation: π2 =

CCD0 ,

the fair coupon is the solution

CCD0 h 0 δ(t) [I0 − exp (Qr,− (xt − ln (θ (π1 + π2 ))))] (R − Q0 )−1 1 (1 − γ)  NC 0 + δ(t) exp (Qr,− (xt − ln (θ (π1 + π2 )))) SC (π2 ) , NS + NC

(59)

where SC (π2 ) is provided by equation (46). π2 From the above corollary, the fair yield of CoCo bonds is then y2 = CCD , which ensures 0 2 sd d CCD0 = D0 . Once that π1 , π1 and π2 satisfy equations (58) and (59), it is easy to check that E0 = S0I .

9

Numerical illustration

Earnings are disclosed at most quarterly and the lack of data prevents using directly the accounting information to t Xt to real time series. Instead, practitioners consider that the daily quoted stock market value is close to a multiple of EBIT. It is thus reasonable to assume that earnings and stock prices are governed by the same dynamics, at least for calibration purposes. Then, based on daily stock quotes of a French bank, Société Générale, a regime-switching process is calibrated by a standard ltering procedure (Hamilton (1989)) and serves later as reference dynamics for Xt . The period considered for the calibration ranges from 2/1/2001 to 10/3/2014 (with 3431 observations). Gatumel and Ielpo (2011) and Guidolin and Timmermann (2007) nd that two regimes are not enough to capture asset dynamics for multiple securities. Their empirical results point out that between two and ve regimes are required to capture the features of each asset's 17

distribution. Based on this observation, models with two to ve regimes are tested and their loglikelihoods, AIC and BIC are presented in Table 2. These statistics show that a model with four states achieves the best t. Table 3 presents drifts and volatilities of stocks return on the annual basis. The matrix of transition probabilities is reported in Table 4. As discussed in Guidolin and Timmermann (2007), each state of δ(t) corresponds to an economic cycle. Table 3 shows that drifts decrease while volatilities increase from State 1 to State 4. So States 1 and 2 are respectively characterized by bull and slow-growth markets, whereas States 3 and 4 are respectively identied as slowing down markets and market crashes. Table 5 shows the average of 12 months Libor rates, observed in each economic regime. It also presents the vector ξ dening the risk neutral measure Q, which is the solution of the system of equations (15). The drifts of Xt under the risk neutral measure Q are obtained from relationship (10). Since µ1 does not satisfy condition (14), the value of µ1 has been lowered from 4.61% to 75% of the risk free rate r1 , i.e. 2.17%. Table 2: Loglikelihoods, AIC N =2 LogLik. 11 446 AIC -22 879 BIC -22 940

and BIC for models N =3 N =4 11 602 11 633 -23 180 -23 225 -23 303 -23 429

Table 3: Drifts, volatilities and standard errors ltered from observations. Estimate Std Err. µ ¯1 7.21% 0.23% µ ¯3 -3.70% 0.87% σ1 6.82% 0.41% σ3 22.09% 1.21%

with 2 to 5 states. N =5 11 630 -23200 -23506

of stocks log-returns, under the real measure P,

µ ¯2 µ ¯4 σ2 σ4

Estimate 6.93% -47.57% 12.85% 41.44%

Std Err. 0.40% 2.01% 0.61% 3.74%

Table 4: Matrix of one-year transition probabilities for δ(t) and standard errors. pi,j (0, 1) state 1 state 2 state 3 state 4 state 1 0.9768 0.0220 0.0005 0.0007 Std Err

state 2 Std Err

state 3 Std Err

state 4 Std Err

0.0427

0.0013

0.0064

0.0002

0.0135

0.9638

0.0224

0.0003

0.0082

0.0521

0.0046

0.0052

0.0005

0.0357

0.9515

0.0123

0.0004

0.0003

0.0342

0.0002

0.0000

0.0015

0.0399

0.9586

0.0007

0.0012

0.0023

0.0861

Table 5: Average 12 M Libor, Esscher vector ξ , and on observations from 2001 to 2014. Libor 12 M Esscher vector, ξ r1 2.89% ξ1 -5.5802 r2 2.43% ξ2 -3.9280 r3 2.38% ξ3 -0.1095 r4 2.88% ξ4 2.2813 18

drift of Xt under Q in each regime, based

µ1 µ2 µ3 µ4

Adjusted EBIT growth, Q 2.17% (4.61%*) 0.44% -4.23% -8.39%

To emphasize interconnections between the capital structure and fair costs of liabilities, the bank is assumed to be founded at time t = 0, when the economy is in State 3 (economic slow down). The initial value x0 of Xt is computed such that the asset value A0 is equal to 100. The purchase of the asset is nanced by equity, CoCo bonds, straight bonds and deposits. Shareholders and depositors both invest 15 when the bank is created. Various allocations between straight and convertible debts are considered, for a total of CCD0 + SD0 = 70, such that the accounting balance sheet is well balanced. Table 6 exhibits other parameters. Table 6: Other parameters. x0 1.2221 λ 50% A0 100 γ 33% E0 15 θ 50% CD0 15 δ(0) 3 NC CCD0 NS E0 Table 7 shows the fair interests computed following procedures detailed in Section 8. First, it is interesting to notice that the total of interests paid, π1 + π2 , decreases slightly with the amount of CoCo bonds, from CCD0 = 65 to 25. The smallest charge of interests is around 5.09 and is greater than the initial earnings, ex0 = 3.39. The income after tax and interest expenses is thus negative just after the creation of the rm. Even if the total interest charge does not vary much, a closer look reveals huge spreads between yields of liabilities, as illustrated in the left sub-gure of Figure 9.1. This graph emphasizes that the higher the volume of CoCo debts is, the lower the cost of straight debts is, and the higher the yields of CoCo bonds and deposits are. These trends are directly related to dierent exposures to default risk. A bank, which is mainly nanced by CoCo bonds, owns a comfortable cushion of capital to absorb potential losses. It has then a low probability of default, contrary to a bank heavily leveraged by straight debts. Later in this paragraph, the analysis of probabilities of bankruptcy will conrm this point. With the risk of bankruptcy reduced in this way, the default risk premium for straight debts is thus small. On the other hand, the deposit insurance is cheap and has a small impact on the yield of deposits, which is closer to the one of straight debts. However, the risk of conversion being high in this scenario, CoCo bonds are well remunerated. Another important factor implied in the C calculation of CoCo rates is the conversion rate NCN+N . A sensitivity analysis to this conversion S ratio concludes this section.

CCD0 65 60 55 50 45 40 35 30 25 20

Table 7: Costs of liabilities per category in percentage and Deposits Straight debts CoCo π1,d SD0 CD0 rate % rate % rate % 5 15 3.32% 4.47% 7.80% 0.4986 10 15 3.26% 4.64% 7.86% 0.4891 15 15 3.20% 4.83% 7.90% 0.4800 20 15 3.14% 5.05% 7.93% 0.4709 25 15 3.08% 5.29% 7.91% 0.4617 30 15 3.02% 5.58% 7.81% 0.4523 35 15 2.95% 5.93% 7.56% 0.4424 40 15 2.87% 6.40% 7.06% 0.4316 45 15 2.79% 7.10% 5.92% 0.4190 50 15 2.65% 8.80% 2.13% 0.3987

absolute value. π1,sd π2 0.2235 0.4644 0.7252 1.0094 1.3227 1.6737 2.0772 2.5613 3.1970 4.3979

5.0724 4.7148 4.3467 3.9628 3.5573 3.1229 2.6494 2.1176 1.4803 0.4267

π1 + π2 5.7945 5.6684 5.5519 5.4431 5.3417 5.2490 5.1689 5.1106 5.0963 5.2232

Table 8 shows the economic value of the rm, which is the sum of all liabilities and equity, minus the deposits insurance, F0 = 100 − DI0 . Any increase of straight debts raises both the probability of default and the cost of deposits insurance. The rm's value is then systematically 19

lower for banks that are highly leveraged by straight debts. In this model, it is clear that deposits insurance represents a friction in the market. Indeed, without this compulsory insurance, the rm's value would be independent from the capital structure, as stated in the Modigliani and Miller's theorem (1958). Table 8 also provides the market value of equity and gross of insurance, in the current economic state (column S0 ) and in other economic regimes, assuming that a transition to these states occurs immediately after the bank's birth. The right sub-gure of Figure 9.1 reveals that except state 3, the equity curves are concave and admit maximum values at the level of straight debts around 45. Liabilities being issued at fair prices in state 3, the shareholders' equity is constant and equal to 15, whatever the capital structure of the bank. Our switching regime model emphasizes that this is no more the case if the economic conjuncture changes. From the shareholder's perspective, there exists an optimal structure of debts maximizing the equity value in economic regimes, dierent from the one in force, during the issuance of debts. This aspect should be taken into consideration for determining the bank's capital structure. More surprisingly, the equity value is slightly higher when the economy is highly depressed (fourth regime) than in slow down mode (third regime). This is explained by the parameters driving Xt in these states, under the risk neutral measure. A quick calculation shows that the expected growth rate of At under Q,   1 2 At E | δ(s) = e3 , ∀s ∈ [0, t] = e(µ3 + 2 σ3 )t = e−1.79% t , A0   1 2 At | δ(s) = e4 , ∀s ∈ [0, t] = e(µ4 + 2 σ4 )t = e0.20% t , E A0 which is greater in state 4 than that in state 3, because the volatility σ4 is important. This means that once nancial markets enter into recession (state 3), economic agents price assets with a very pessimistic set of assumptions. Whereas when the economy hits rock bottom (state 4), assets are valued with an assumption of high volatility and of soft growth4 .

Straight Debts SD0 5 10 15 20 25 30 35 40 45 50

Table 8: Firm values, and equity values. I , Equity Insurance Firm Value S0+ S0 , δ(0) = e3 DI0 F0 δ(0+) = e1 18.8450 3.8449 96.1551 70.4433 19.4680 4.4678 95.5322 73.7005 20.0725 5.0724 94.9276 76.5372 20.6708 5.6706 94.3294 79.0631 21.2735 6.2734 93.7266 81.3247 21.8928 6.8927 93.1073 83.3196 22.5452 7.5450 92.4550 84.9815 23.2591 8.2590 91.7410 86.1309 24.1029 9.1028 90.8972 86.3038 25.4674 10.4673 89.5327 83.4463

4

I , S0+ δ(0+) = e2 33.9132 35.4458 36.8432 38.1159 39.2546 40.2284 40.9728 41.3633 41.1387 39.4691

I , S0+ δ(0+) = e4 16.0851 16.6480 17.2506 17.8903 18.5618 19.2551 19.9482 20.5899 21.0391 20.4754

Decreasing the volatility σ4 from 41.44% to 31.44% , leads to an asset growth rate equal to µ4 + 12 σ42 = 3.45% and for this volatility, the equity is lower in state 4 than in state 3.


20

CoCo Bonds 60

50

CoCo Bonds

40

30

20

8

100

8

7

30

20 100

90

90

80

80

5

5

4

4

3

3

40

Equity s1 Equity s2 Equity s3 Equity s4

60

%

6

30 Straight Debts

40

70

6

20

50

7

Deposits rate Straight Debts rate CoCo bonds rate

10

60

60

50

50

40

40

30

30

20

20

10

10

0

50

70

10

20

30 Straight Debts

40

0 50

Figure 9.1: Cost of liabilities and Equity in each economic regime. Although nancing by a contingent convertible debt is an ecient way for a bank to mitigate the bankruptcy risk, it introduces a risk of dilution. In case of conversion of a massive convertible debt, former shareholders can avoid bankruptcy at the cost of losing control of the rm. In Section 7, indicators have been introduced to monitor these risks. Figure 9.2 plots the expected time before conversion against the average lifetime before ruin for several allocations of liabilities. In the particular case studied, the conversion occurs in average between 14 to 27 years, whereas the expected life time varies between 33 and 240 years. Here, expected times are computed under the risk neutral measure Q and earnings grow slower than the risk-free rate. Replacing risk neutral drifts by historical returns would immediately modify the value of these indicators. A bank highly leveraged by CoCo bonds has a longer expected lifetime and a lower probability of bankruptcy than a rm mainly nanced by straight debts. But the shareholders' voting rights are diluted on average after a short period of time. This observation is conrmed by Figure 9.3, which presents the probabilities of bankruptcy and conversion from 1 to 50 years. 28

Expected time before CoCo conversion

(30 CC,40 SD) 26 (20 CC,50 SD) (40 CC,30 SD)

24

22 (50 CC,20 SD) 20

18 (60 CC,10 SD) 16

14

0

50

100 150 Expected time before ruin

200

250

Figure 9.2: Expected time before ruin against average time before CoCo conversion.

21

0.03

0.4

0.35 0.025

Probability of Ruin

Probability of Conversion

(65 CC,5 SD) (45 CC,25 SD) (25 CC,45 SD)

0.02

(65 CC,5 SD) (45 CC,25 SD) (25 CC,45 SD)

0.3

0.015

0.01

0.25

0.2

0.15

0.1 0.005 0.05

0

10

20 30 Time horizon

40

0

50

10

20 30 Time horizon

40

50

Figure 9.3: Probabilities of conversion and bankruptcy, per time horizon. Table 9 analyzes the sensitivity of the fair cost of contingent debts to the conversion ratio, and shows that CoCo bonds with a higher conversion ratio are cheaper than those with a lower one. Intuitively, a lower conversion rate gives the right to a smaller fraction of earnings after conversion. To compensate this shortfall, the fair coupon paid before conversion is higher. The left graph of Figure 9.4 illustrates that the CoCo coupon declines monotonically with its conversion rate. And the right graph of the same gure shows the probabilities of swapping the CoCo bond to equity for dierent conversion rates. Paradoxically, for higher conversion ratios, the probabilities are signicantly lower and the expected time before the swap is also longer. This is in fact a direct consequence of lower CoCo rates. If the bank issues CoCo with a high conversion ratio, the immediate total interest charge π1 + π2 is low and the average delay before swapping the convertible into equity is lengthened. Probabilities of default are not reported because the likelihood of a default is not aected by the conversion rate in this extreme case. NC NC +NS

Table 9: Impact on conversion rate on pricing of CoCo bonds. SD0 = 30, CCD0 = 40. π2 C Conversion rate, NCN+N CoCo rate, CCD , E (τ1 | F0 ) P (τ1 ≤ 10 years) 0 S 0.6500 10.31% 7.11 y 18.62% 0.7000 8.34% 20.10 y 1.84% 0.7500 7.45% 27.03 y 0.74% 0.8000 6.85% 31.97 y 0.40% 0.8500 6.38% 36.41 y 0.25% 0.9000 6.00% 39.91 y 0.17%

22

0.25 NC/(NC+NS)=0.7

10

NC/(NC+NS)=0.8

CoCo bonds rate

0.2 Probability of Conversion

9.5 9

%

8.5 8 7.5 7

NC/(NC+NS)=0.9

0.15

0.1

0.05

6.5 6 0.65

0.7 0.75 0.8 0.85 Conversion ratio NC/(NC+NS)

0

0.9

0

10

20 30 Time horizon

40

50

Figure 9.4: Probabilities of conversion and bankruptcy, per time horizon. 10

Conclusions

This paper emphasizes the link between the capital structure of a bank and the fair costs of its liabilities, when the operating prot is aected by variations of macroeconomic conditions. Contingent convertibles automatically recapitalize the nancial institution in dicult times and are thus ecient instruments to mitigate the risk of bankruptcy. Furthermore, numerical analysis shows that a high volume of convertible debts reduces the yield oered to straight debts. When the probability of default is lower, the cost of deposits insurance falls and depositors in counterpart receive a higher compensation. But it does not totally removed the risk of bankruptcy. On the other hand, the risk of dilution is huge for shareholders. However, it is rather surprising that this risk can be mitigated by increasing the conversion rate of the convertible debt. If this debt is fairly priced, the high conversion rate is compensated by a low interest cost before conversion. The total nancial charge and the probability that earnings fall below the regulatory threshold is reduced. This postpones on average the conversion of CoCo bonds into equity. By construction, when liabilities are issued at fair prices, the shareholders' equity has a constant market value, whatever the capital structure of the bank. The switching regime model emphasizes that this is no more the case if the economic conjuncture changes. From the shareholder's perspective, there exists an optimal structure of debts maximizing the equity value in economic regimes, dierent from the one in force, during the issuance of debts. This aspect should be taken into consideration for optimizing the bank's capital structure.

References

[1] Abate J. , Whitt. W. 2006. "A Unied Framework for Numerically Inverting Laplace Transforms." INFORMS Journal of Computing, 18.4 , 408-421. [2] Ammann M. and Genser M. 2005. Making Structural Credit Risk Models Testable: Introducing Complex Capital Structures. University of St. Gallen, Working paper. [3] Altman E. Resti A. Sironi A. 2002. The link between default and recovery rates: eects on the procyclicality of regulatory capital ratios. BIS Working Paper, 113.

23

[4] Barucci E. Del Viva L. 2012. Countercyclical contigent capital. Journal of Banking & Finance, 36, 1688-1709. [5] Barucci E. Del Viva L. 2013. Dynamic capital structure and the contingent capital option. Annals of Finance, 9 (3), 337-364. [6] Bradley M., Jarrell G.A , Kim E.H. 1984. On the existence of an optimal capital structure: theory and evidence. Journal of Finance, 39 (3), 857-878. [7] Brennan, M., Schwartz E. 1978, Corporate income taxes, valuation, and the problem of optimal capital structure, Journal of Business 51, 103-114. [8] Bungton J., Elliott R.J. 2002. American options with regime switching. International Journal of Theoretical and Applied Finance 5, 497-514. [9] Calvet L., Fisher A. 2001. Forecasting multifractal volatility. Journal of econometrics. 105, 17-58. [10] Calvet L., Fisher A. 2004. How to forecast long-Run volatility: regime switching and the estimation of multifractal processes. Journal of Financial Economics 2, 49-83. [11] Chen N. Kou S. 2009. Credit spreads, optimal capital structure and implied volatility with endogeneous default and jump risk. Mathematical Finance, 19 (3), 343378. [12] Cholette L., Heinen A., Valdesogo Al. 2009. Modelling international nancial returns with a multivariate regime switching copula. Journal of nancial econometrics 7 (4), 437-480. [13] Corcuera J.M., De Spiegeleer J., Fajardo J., Jönsson H., Schoutens W., Valdivia A. Close form pricing formulas for Coupon Cancellable CoCos. Forthcoming Journal of Banking & Finance, doi: http:// dx.doi.org/10.1016/j.jbankn.2014.01.025 . [14] De Spiegeleer J., Schoutens W. 2012. Steering a bank around a death spiral: Multiple Trigger CoCos. Wilmott Magazine 59, 62-69. [15] De Spiegeleer J., Schoutens W. 2013. Multiple Trigger CoCos: Contingent debt without death-spiral risk. Financial Markets, Institution and Instruments Journal 22(2), 129-141. [16] Elliott R. J., Chan L., Siu T. K. 2005. Option pricing and Esscher transform under regime switching. Annals of Finance 1, 423-432. [17] Elliott R. J., Siu T. K. 2013. Option pricing and ltering with hidden Markov-modulated pure-jump processes. Applied Mathematical Finance 20(1), 1-25. [18] Gatumel M., Ielpo F. 2011. The Number of Regimes Across Asset Returns: Identication and Economic Value. Working paper, available at SSRN 1925058. [19] Gerber H.U., Shiu E.S.W. 1994. Options Pricing by Esscher Transform. Transactions of the society of actuaries, 26, 99-191. [20] Glasserman P., Nouri B. 2012. Contingent Capital with a Capital-Ratio Trigger. Management Science,58 (10), 1816-1833 [21] Guidolin M., Timmermann A. 2005. Economic Implications of bull and bear regimes in UK stock and bond returns. The Economic Journal, 115, 11-143 [22] Guidolin M., Timmermann A. 2007. Asset allocation under multivariate regime switching. Journal of Economic Dynamics and Control, 31 (11), 3503-3544. 24

[23] Guidolin M., Timmermann A. 2008. International Asset Allocation under Regime Switching, Skew, and Kurtosis Preferences. Review of Financial Studies. 21 (2), 889-935. [24] Hackbarth D., Miao J., Morellec E. 2006. Capital structure, credit risk and macroeconomics conditions. Journal of Financial Economics. 82, 519-550. [25] Hainaut D., MacGilchrist R. 2012 Strategic asset allocation with switching dependence. Annals of Finance 8 (1), 75-96. [26] Hamilton J.D. 1989 "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle." Econometrica. 57 (2), 357-384. [27] Harding J., Liang X., Ross S.L. 2007. The optimal capital structure of banks: balancing deposit insurance, capital requirements and tax-advantaged debt. Working paper University of Connecticut. [28] Hilberink B., Rogers L.C.G. 2002. Optimal capital structure and endogenous default. Finance and stochastics 6, 237-263. [29] Jiang Z., Pistorius M.R. 2008. On perpetual American put valuation and rst-passage in a regime-switching model with jumps. Finance and Stochastics, 12 (3), 331-355. [30] Koziol C., Lawrenz J. 2012. Contingent convertibles. Solving or seeding the next banking crisis. Journal of Banking & Finance, 36, 90-104. [31] Leland H. 1994. Corporate debt value, bond covenants, and optimal capital structure. Journal of Finance 49, 12131252. [32] Leland, H., and K. Toft, 1996, Optimal Capital Structure, Endogenous Bankruptcy, and the Term Structure of Credit Spreads, Journal of Finance 51, 987-1019. [33] Leland H., 1998, Agency Costs, Risk Management, and Capital Structure, Journal of Finance 53, 1213-1243. [34] Myers S. C. 1984. The Capital Structure Puzzle. Journal of Finance 39 (3), 575-592. [35] Modigliani F., Miller M. 1958. The Cost of Capital, Corporation Finance and the Theory of Investment, American Economic Review 48, 261297. [36] Pennacchi G. 2010. A structural model of contingent bank capital. The federal Reserve Bank of Cleveland. Working paper. [37] Rogers L., Shi Z. 1994. Computing the invariant law of a uid model. Journal of Applied Probability, 31 (4), 885896. [38] Titman S., Wessels R. 1988. The determinants of capital structure choice. Journal of Finance 43 (1), 1-19. [39] Titman S., Tsyplakov S. 2007. A Dynamic Model of Optimal Capital Structure. Review of Finance 11 (3), 401-451.

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